Statement

Any Normal subgroup (?) of a group that is self-centralizing, is also normality-large: its intersection with every nontrivial normal subgroup is nontrivial.

Proof

Given: A group , a normal subgroup , such that .

To prove: If is a nontrivial normal subgroup of such that is trivial, then is trivial.

Proof: Consider the commutator. This is contained both in and in , since they are both normal. Hence it is contained in , which is trivial. Thus, every element of commutes with every element of , so . Since is self-centralizing, we get . Since is trivial, we get that is trivial.